1. Problems Involving Linear Systems
Section 3.6
Problems Involving Linear Systems

2. I) Number of Solutions:x y -4 -3 -2 -1 1 2 3 4
1. NO SOLUTIONS
(System is Inconsistent)
Lines are parallel and do not intersect each other
Lines have the same slope but different y-intercept x y -4 -3 -2 -1 1 2 3 4
2. ONE SOLUTION
(System is Consistent)
Lines intersect at ONE point
Lines have Different slopes
3. INFINITE SOLUTIONS
(System is Consistent) x y -4 -3 -2 -1 1 2 3 4
Lines OVERLAP each other
Lines have both the SAME slope and Y-intersect

3. Ex: Indicate how many solutions are in each system
Slopes are different
 One Intersection
Slopes are the same
 No Intersections
Y-intercepts are different
Slopes are the same
Y-intercepts are the same
 Infinite Solutions

4. II) Number of Solutions in Standard Form:
A Linear System with INFINITE Solutions will have all 3 coefficients A,B,C in ratio with the same constant
All corresponding coefficients are in ratio and 4 times bigger
Same Slopes and Same Y-intercept
A Linear System with NO Solutions will only have coefficients A & B in ratio with the same constant and NOT “C”
Only coefficients A & B are in ratio and not “C”
Same Slopes but Different Y-intercepts
A Linear System with ONE Solution will have different ratios for coefficients A & B. “C” doesn’t matter.

5. Ex: Indicate the Number of Solutions in each Linear System:
All coefficients A,B,C are in ratio
Coefficients A & B - NOT in ratio
Infinite Solutions
One Solutions
Coefficients A & B are NOT in ratio
A & B are in ratio, but not “C”
One Solutions
NO Solutions

6. III) Solving Problems Involving Linear Systems
In the next part, you will have scenarios that involves problems with linear equations
First step: Indicate What the Variables are
Number of People
Cost for a certain item
2nd Step: Read the Question to generate your 2 equations
Revenue
Interest Earned:
3rd Step: Solve the system by Elimination or Substitution
4th Step: Write your concluding statement

7. Ex: A tutoring center charges an annual fee and an hourly fee
Ex: A tutoring center charges an annual fee and an hourly fee. 8 hours of tutoring cost $ hours cost $500. Find the annual cost and hourly cost.
Let “x” be the Annual Cost
1st: Indicate the Variables
Let “y” be the Hourly Cost
2nd: Make the Equations
Solve by Elimination
The Hourly cost is $30 per hour
The Annual cost is $50 per year

8. Practice: The cost for a school play is $35 per adult and $20 per student. 160 people attended the play and total revenue was $ How many students and adults attended?
1st: Indicate the Variables
Let “x” be the Number of Adults
2nd: Make the Equations
Let “y” be the Number of Students
Quantity
Revenue
3rd: Solve by Elimination
100 students and 60 parents attended the school play

9. Ex: James invested $9000, part with Bank A (3%) and part with Bank B (5%). After one year, he made a total of $340 in interest. How much did he invest with each bank?
Total Investment $9000
Indicate the Variables
Let “A” be amount invested in Bank A
Amount $A
Bank A (3%)
Amount $B Bank B (5%)
Let “B” be amount invested in Bank B
James invested $5500 with Bank A and $3500 with Bank B
Make the Equations
Solve by Elimination

10. Indicate the Variables
Ex: James invested $10000, part with Bank A (7%) and part with Bank B (13%). After one year, both banks made the same amount of Interest. How much did he invest with each bank?
Total Investment $10000
Indicate the Variables
Let “A” be amount invested in Bank A
Amount $A
Bank A (7%)
Amount $B Bank B (13%)
Let “B” be amount invested in Bank B
Make the Equations
These are the interests he earned from each bank
Solve by Substitution
James invested $3500 with Bank A and $6500 with Bank B

11. Challenge: A musical charges $4. 00 for adults and $2. 50 for children
Challenge: A musical charges $4.00 for adults and $2.50 for children. On the first night, the ratio of adults to kids was 3:5. On the second night, the ratio was 2:3. A total of 1390 people attended for two nights, and the revenue generated was $ How many adults and kids attended in each night?
Adults $4.00
Kids $2.50
Total Attendance 3x
1st Night 5x 8x
2nd Night 2y 3y 5y
Revenue
4 x (3x+2y)
2.5 x(5x+3y)

12. Challenge: A musical charges $4. 00 for adults and $2. 50 for children
Challenge: A musical charges $4.00 for adults and $2.50 for children. On the first night, the ratio of adults to kids was 3:5. On the second night, the ratio was 2:3. A total of 1390 people attended for two nights, and the revenue generated was $ How many adults and kids attended?
Make the Equations
Quantity
Revenue

13. Challenge: A musical charges $4. 00 for adults and $2. 50 for children
Challenge: A musical charges $4.00 for adults and $2.50 for children. On the first night, the ratio of adults to kids was 3:5. On the second night, the ratio was 2:3. A total of 1390 people attended for two nights, and the revenue generated was $ How many adults and kids attended?
Adults $4.00
Kids $2.50
Total Attendance
3(80)
5(80)
1st Night
=240
=400
2(150)
3(150)
2nd Night
=300
=450
Total Attended
=540 adults
=850 kids